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{8} • Numerical Methods There’s nothing special about the interval [0,1] either. We could replace [0,1] with any interval [a,b]. Provided f is positive at one end and negative at the other end, then f (x) = 0 must have a solution in [a,b]. Finally, there’s also nothing special about the value 0. For any real number value c, if f < c at one end of the interval [a,b], and f > c at the other end, then there must exist an x ∈ [a,b] such that f (x) = c. We summarise this discussion with a statement of the intermediate value theorem. Theorem (Intermediate Value Theorem) Let f : [a,b] → R be a continuous function, and c be a real number. a If f (a) < c and f (b) > c, then there exists an x ∈ [a,b] such that f (x) = c. b If f (a) > c and f (b) < c, then there exists an x ∈ [a,b] such that f (x) = c. Note that the intermediate value theorem doesn’t say anything about how many times f (x) takes the value c. There might be many values of x in the interval [a,b] such that f (x) = c. All the theorem says is that there is at least one. If you’re on one side of the river, and later you’re on the other side of the river, then you must have crossed the river — you might have crossed it many times, but you certainly crossed it at least once! Bisection algorithm We now describe the bisection algorithm in detail. Suppose we have a continuous function f : [a,b] → R, and we want to solve f (x) = 0. (To solve for some other value of f , i.e. f (x) = c, you can rearrange the equation to f (x) − c = 0.) We assume f (a) and f (b) are both nonzero — otherwise we already have a solution! We also assume f (a) and f (b) have opposite signs. The intermediate value theorem then ensures that f (x) = 0 has a solution for some x ∈ [a,b]. As we have seen, the idea is to look at the value of f (x) at the midpoint a+b 2 of the interval [a,b], i.e. the average (or mean) of a and b. If f ( a+b 2 a+b ) = 0, then we have a solution, and we can stop. Otherwise, f ( 2 ) is either ) and f (b), the intermediate positive or negative. Depending on the signs of f (a), f ( a+b 2 value theorem will tell us that one or the other of the intervals [ a, a + b 2 ] or [ a + b 2 ,b ] contains a solution. We call this interval [a 1 ,b 1 ].
A guide for teachers – Years 11 and 12 • {9} We then repeat the process, evaluating f at the midpoint a 1+b 1 2 of [a 1 ,b 1 ]. If f we have a solution, and stop. Otherwise, f ( a1 +b 1 2 ( ) a1 +b 1 2 = 0, ) ≠ 0 and the intermediate value theorem tells us that one or the other of the two sub-intervals obtained by bisecting [a 1 ,b 1 ] contains a solution. We call this interval [a 2 ,b 2 ]. We continue in this fashion, obtaining intervals [a 3 ,b 3 ], [a 4 ,b 4 ], and so on. We now state the bisection method rigorously. [Bisection method] Let f : [a,b] → R be a continuous function with f (a) and f (b) both nonzero, and of opposite sign. We seek a solution of f (x) = 0. Let [a 0 ,b 0 ] = [a,b]. At the n’th step of the method, we have an interval [a n−1 ,b n−1 ], where f (a n−1 ) and f (b n−1 ) are both nonzero and of opposite signs. Then: 1 Calculate f 2 Otherwise, f • f • f ( an−1 +b n−1 2 ( an−1 +b n−1 2 ) ( ) . If f an−1 +b n−1 2 = 0, x = a n−1+b n−1 2 is a solution, and we stop. ) ≠ 0. Precisely one of the following two possibilities occurs: ) [ and f (a n−1 ) have opposite signs. Then let [a n ,b n ] = a n−1 , a n−1+b n−1 2 ( an−1 +b n−1 2 ( an−1 +b n−1 2 ) and f (b n−1 ) have opposite signs. Then let [a n ,b n ] = [ an−1 +b n−1 2 ,b n−1 ]. 3 If the interval [a n ,b n ] has the desired degree of accuracy for a solution, stop. Otherwise, return to step 1 and continue. Note that the intervals [a n ,b n ] produced by the bisection method are all nested: [a 0 ,b 0 ] ⊃ [a 1 ,b 1 ] ⊃ [a 2 ,b 2 ] ⊃ ··· . Moreover, each of these intervals is half as long as the previous one: [a n ,b n ] is half the length of [a n−1 ,b n−1 ]. Exercise 2 How many times smaller is the length of the interval [a n ,b n ] compared to the original [a,b] = [a 0 ,b 0 ]? In other words, what is b n − a n b − a ? ] . Using the bisection algorithm Let’s now turn to some examples. We’ll obtain some approximations to some well-known irrational numbers using the bisection method. Example Find π to within an accuracy of 0.05 by approximating the solution to sin x = 0 in the interval [3,4] using the bisection method.
Page 1 and 2: VCAA-AMSI maths modules 348 9012 x
Page 3 and 4: Assumed knowledge . . . . . . . . .
Page 5 and 6: A guide for teachers - Years 11 and
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